Calculus 3-4 Rates of Change

Instantaneous Rate of Change 𝑓 ′ 𝑥 The derivative at a given point will tell you the instantaneous rate of change (slope) at that point

𝐴=𝜋 𝑟 2 Find 𝑑𝐴 𝑑𝑟 when 𝑟=2, 𝑎𝑛𝑑 𝑟=5 𝑑𝐴 𝑑𝑟 =2𝜋𝑟 𝑟=2: 𝑑𝐴 𝑑𝑟 =4𝜋 𝑟=2: 𝑑𝐴 𝑑𝑟 =4𝜋 𝑟=5: 𝑑𝐴 𝑑𝑟 =10𝜋 Why is 𝑑𝐴 𝑑𝑟 larger at 𝑟=5 As the radius increase the rate that the area of circle increase as well

Linear Motion 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦= 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ′ 𝑠 𝑡 =4 𝑥 2 +8𝑥−9 𝑣 𝑡 =8𝑥+8

Motion under Gravity 𝑠 𝑡 = 𝑠 0 + 𝑣 0 𝑡− 1 2 𝑔 𝑡 2 𝑣 𝑡 = 𝑑𝑠 𝑑𝑡 = 𝑣 0 −𝑔𝑡 𝑔≈9.8 𝑚 𝑠 2 𝑜𝑟 32 𝑓𝑡 𝑠 2

Find the velocity of an object dropped from a height of 300 m at the moment it his the ground. Equation for the object How long before it hits the ground Velocity function Velocity at that time

𝑠 𝑡 =300− 1 2 9.8 𝑡 2 𝑡=7.8246 𝑠𝑒𝑐𝑠 𝑣 𝑡 =−9.8𝑡 𝑣 0 =−76.68108 𝑚/𝑠

Problems 3.4 #1, 7, 9-11, 15-37 odd NOT 23 and 29, 38, and 49